On the Eigensystems of Graded Matrices

Abstract

Informally a graded matrix is one whose elements show a systematic decrease or increase as one passes across the matrix. It is well known that graded matrices often have small eigenvalues that are determined to high relative accuracy. Similarly, the eigenvectors can have small components that are nonetheless well determined. In this paper, we give approximations to the eigenvalues and eigenvectors of a graded matrix in terms of a base matrix that show how these phenomena come about. This approach provides condition numbers for eigenvalues and individual components of the eigenvectors. The results are applied to derive related results for the singular value decomposition. (Also cross-referenced as UMAICS-TR-2000-01)